The beauty of future editions of a textbook is that errors can beremoved and improvement of teaching can be improved. Already I made amajor error, in that suggesting that we do not connect points of thegraph of the function when we have holes between points, for wecertainly do connect them since the hypotenuse atop the picketfence isa straight line segment that is connected to a "number point of thegraph". In fact, that is what the Fundamental Theorem of Calculus isgoing to be in New Math. In Old Math, they thought their importantFundamental theorem was going to be the union of the derivative asinverse to the integral. In New Math we build the derivative asinverse and why need to prove it when we built it as such. So theFundamental Theorem of Calculus in New Math concerns itself with whyand how is it that Euclidean geometry with straight lines and straightline segments is the only geometry that can build a Calculus, whileElliptic geometry and Hyperbolic geometry are impossible geometries tobuild a Calculus therein?

Now I realized after post-page #3 yesterday (almost 1/3 done with thebook) that it is too complicated for a High School student. So thispost, which should be #4 is now numbered to be #0 so as to make simpleand inviting to High School students. This should be the first page sothat I can constantly refer to this simplified model.

Exercise on Graph Paper

Now I do not know if I can get 100 dots per line in the Usenetsci.math newsgroup. Let me try.

The above is a grid of 100 dots for the x-axis and 100 dots for the y-axis and 100x100=10,000 dots altogether. I doubt the post will show itin a square 100 dots wide and 100 dots long.

So I need the student or reader to get a graph pad. I have anEngineering graph pad where each sheet is 35 dots wide and 50 dotslong so I need to take 6 sheets and cut and paste, or cut and tapethem together to form a large sheet that is 100 dots wide and 100 dotslong for a total of 10,000 dots altogether. So if a High Schoolstudent, it is wise to do this exercise for you want to refer to itand to constantly make pencil drawings of functions and graphs to knowwhat is going on.

Now we pretend, pretend that 10 is infinity borderline, the borderlinewhere 10 is the last finite number and the largest finite number. Sothat has a deep and powerful implication. For it means that 1/10 isalso a borderline of the small.

Now mathematics is the science of precision and that means thatmathematics can handle precision and accuracy only with finiteentities, finite numbers, finite points of geometry. Mathematicsstarts deteriorating once it reaches infinity, for it loses precisionand accuracy. When mathematics is in infinity territory, it no longeris mathematics. The planet Earth is a good analogy here. The planetEarth has a borderline of solid matter Earth which is the surface ofEarth as a borderline, including water of oceans. Another borderlineis the atmosphere, but beyond the atmosphere we start getting intoOuter Space. Now the magnetosphere of Earth that protects Earth fromSolar rays that are harmful can be considered a borderline for Earthbut beyond that, it is hard to say that Earth has any physicalparameters. Now that is an analogy to mathematics. Mathematics is thescience of precision and can be precise about finite numbers andfinite points of geometry but once it reaches the borderline ofinfinity numbers or infinity points of geometry, we no longer havemathematics of total precision, and math begins to fall apart intoimprecision. Now in the next post, I will talk about the preciseborderline of infinity, but in this post, we pretend the number 10 isthat infinity borderline. For High School students, they can handlethe number 0, 1/10, 2/10, 3/10, .. on up to 10. And they can easilygraph functions with only these numbers in play. They can drawpicketfence structures on this large graph paper.

And as I write the next 9 pages of this 10 page textbook, I willconstantly refer to this 100 x 100 = 10,000 points of the CartesianCoordinate System.

--More than 90 percent of AP's posts are missing in the Googlenewsgroups author search archive from May 2012 to May 2013. DrexelUniversity's Math Forum has done a far better job and many of thosemissing Google posts can be seen here: